LMFDB - Embedded newform 207.6.c.a.206.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(206,207)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("207.206");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			206.2
Character	$\chi$	$=$	207.206
Dual form			207.6.c.a.206.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.22736i q^{2} -6.78003 q^{4} -105.915 q^{5} +86.6285i q^{7} +157.054i q^{8} +O(q^{10})$	\(q+6.22736i q^{2} -6.78003 q^{4} -105.915 q^{5} +86.6285i q^{7} +157.054i q^{8} -659.570i q^{10} -110.356 q^{11} -288.381 q^{13} -539.467 q^{14} -1194.99 q^{16} -432.456 q^{17} -503.410i q^{19} +718.106 q^{20} -687.224i q^{22} +(-2081.31 + 1450.69i) q^{23} +8092.95 q^{25} -1795.85i q^{26} -587.344i q^{28} -3398.45i q^{29} +7196.66 q^{31} -2415.93i q^{32} -2693.06i q^{34} -9175.25i q^{35} -6315.03i q^{37} +3134.92 q^{38} -16634.3i q^{40} +2359.38i q^{41} -1425.67i q^{43} +748.215 q^{44} +(-9033.97 - 12961.1i) q^{46} -12088.2i q^{47} +9302.49 q^{49} +50397.7i q^{50} +1955.23 q^{52} +16009.7 q^{53} +11688.3 q^{55} -13605.3 q^{56} +21163.4 q^{58} +30063.4i q^{59} -13761.4i q^{61} +44816.2i q^{62} -23194.9 q^{64} +30543.8 q^{65} -19961.9i q^{67} +2932.07 q^{68} +57137.6 q^{70} -59198.6i q^{71} -42846.2 q^{73} +39325.9 q^{74} +3413.14i q^{76} -9559.95i q^{77} -12833.0i q^{79} +126567. q^{80} -14692.7 q^{82} +100089. q^{83} +45803.5 q^{85} +8878.17 q^{86} -17331.8i q^{88} -65919.5 q^{89} -24982.0i q^{91} +(14111.3 - 9835.72i) q^{92} +75277.7 q^{94} +53318.6i q^{95} +127964. i q^{97} +57930.0i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 600 q^{4}+O(q^{10}) $ 40 * q - 600 * q^4	$ 40 q - 600 q^{4} - 1048 q^{13} + 9728 q^{16} + 14704 q^{25} + 4640 q^{31} - 91864 q^{46} - 8192 q^{49} + 150360 q^{52} + 134592 q^{55} - 195704 q^{58} - 183416 q^{64} - 257448 q^{70} + 31088 q^{73} - 77096 q^{82} - 368760 q^{85} - 123512 q^{94}+O(q^{100}) $ 40 * q - 600 * q^4 - 1048 * q^13 + 9728 * q^16 + 14704 * q^25 + 4640 * q^31 - 91864 * q^46 - 8192 * q^49 + 150360 * q^52 + 134592 * q^55 - 195704 * q^58 - 183416 * q^64 - 257448 * q^70 + 31088 * q^73 - 77096 * q^82 - 368760 * q^85 - 123512 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/207\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			6.22736i			1.10085i	0.834884	+	0.550426i	$0.185535\pi$
$2$			6.22736i			1.10085i	−0.834884	+	0.550426i	$0.814465\pi$
$3$		0			0
$3$		0			0
$4$	−6.78003			−0.211876
$5$	−105.915			−1.89466			−0.947331	−	0.320256i	$-0.896231\pi$
$5$	−105.915			−1.89466			−0.947331	+	0.320256i	$0.896231\pi$
$6$		0			0
$7$			86.6285i			0.668215i	0.942535	+	0.334107i	$0.108435\pi$
$7$			86.6285i			0.668215i	−0.942535	+	0.334107i	$0.891565\pi$
$8$			157.054i			0.867608i
$9$		0			0
$10$		−	659.570i		−	2.08574i
$11$	−110.356			−0.274987			−0.137494	−	0.990503i	$-0.543905\pi$
$11$	−110.356			−0.274987			−0.137494	+	0.990503i	$0.543905\pi$
$12$		0			0
$13$	−288.381			−0.473269			−0.236635	−	0.971599i	$-0.576044\pi$
$13$	−288.381			−0.473269			−0.236635	+	0.971599i	$0.576044\pi$
$14$	−539.467			−0.735606
$15$		0			0
$16$	−1194.99			−1.16698
$17$	−432.456			−0.362927			−0.181464	−	0.983398i	$-0.558083\pi$
$17$	−432.456			−0.362927			−0.181464	+	0.983398i	$0.558083\pi$
$18$		0			0
$19$		−	503.410i		−	0.319917i	−0.987124	−	0.159959i	$-0.948864\pi$
$19$		−	503.410i		−	0.319917i	0.987124	−	0.159959i	$-0.0511361\pi$
$20$	718.106			0.401433
$21$		0			0
$22$		−	687.224i		−	0.302721i
$23$	−2081.31	+	1450.69i	−0.820383	+	0.571814i
$23$	−2081.31	+	1450.69i	−0.820383	+	0.571814i
$24$		0			0
$25$	8092.95			2.58974
$26$		−	1795.85i		−	0.521000i
$27$		0			0
$28$		−	587.344i		−	0.141579i
$29$		−	3398.45i		−	0.750389i	−0.926946	−	0.375194i	$-0.877576\pi$
$29$		−	3398.45i		−	0.750389i	0.926946	−	0.375194i	$-0.122424\pi$
$30$		0			0
$31$	7196.66			1.34501			0.672507	−	0.740091i	$-0.265218\pi$
$31$	7196.66			1.34501			0.672507	+	0.740091i	$0.265218\pi$
$32$		−	2415.93i		−	0.417070i
$33$		0			0
$34$		−	2693.06i		−	0.399530i
$35$		−	9175.25i		−	1.26604i
$36$		0			0
$37$		−	6315.03i		−	0.758352i	−0.925325	−	0.379176i	$-0.876208\pi$
$37$		−	6315.03i		−	0.758352i	0.925325	−	0.379176i	$-0.123792\pi$
$38$	3134.92			0.352182
$39$		0			0
$40$		−	16634.3i		−	1.64382i
$41$			2359.38i			0.219199i	0.993976	+	0.109599i	$0.0349568\pi$
$41$			2359.38i			0.219199i	−0.993976	+	0.109599i	$0.965043\pi$
$42$		0			0
$43$		−	1425.67i		−	0.117584i	−0.998270	−	0.0587920i	$-0.981275\pi$
$43$		−	1425.67i		−	0.117584i	0.998270	−	0.0587920i	$-0.0187249\pi$
$44$	748.215			0.0582632
$45$		0			0
$46$	−9033.97	−	12961.1i	−0.629483	−	0.903121i
$47$		−	12088.2i		−	0.798210i	−0.916905	−	0.399105i	$-0.869321\pi$
$47$		−	12088.2i		−	0.798210i	0.916905	−	0.399105i	$-0.130679\pi$
$48$		0			0
$49$	9302.49			0.553489
$50$			50397.7i			2.85093i
$51$		0			0
$52$	1955.23			0.100274
$53$	16009.7			0.782878			0.391439	−	0.920204i	$-0.371977\pi$
$53$	16009.7			0.782878			0.391439	+	0.920204i	$0.371977\pi$
$54$		0			0
$55$	11688.3			0.521008
$56$	−13605.3			−0.579748
$57$		0			0
$58$	21163.4			0.826067
$59$			30063.4i			1.12437i	0.827012	+	0.562184i	$0.190039\pi$
$59$			30063.4i			1.12437i	−0.827012	+	0.562184i	$0.809961\pi$
$60$		0			0
$61$		−	13761.4i		−	0.473521i	−0.971568	−	0.236760i	$-0.923914\pi$
$61$		−	13761.4i		−	0.473521i	0.971568	−	0.236760i	$-0.0760857\pi$
$62$			44816.2i			1.48066i
$63$		0			0
$64$	−23194.9			−0.707852
$65$	30543.8			0.896685
$66$		0			0
$67$		−	19961.9i		−	0.543270i	−0.962400	−	0.271635i	$-0.912436\pi$
$67$		−	19961.9i		−	0.543270i	0.962400	−	0.271635i	$-0.0875643\pi$
$68$	2932.07			0.0768956
$69$		0			0
$70$	57137.6			1.39372
$71$		−	59198.6i		−	1.39369i	−0.717223	−	0.696844i	$-0.754587\pi$
$71$		−	59198.6i		−	1.39369i	0.717223	−	0.696844i	$-0.245413\pi$
$72$		0			0
$73$	−42846.2			−0.941034			−0.470517	−	0.882391i	$-0.655933\pi$
$73$	−42846.2			−0.941034			−0.470517	+	0.882391i	$0.655933\pi$
$74$	39325.9			0.834834
$75$		0			0
$76$			3413.14i			0.0677828i
$77$		−	9559.95i		−	0.183751i
$78$		0			0
$79$		−	12833.0i		−	0.231346i	−0.993287	−	0.115673i	$-0.963098\pi$
$79$		−	12833.0i		−	0.231346i	0.993287	−	0.115673i	$-0.0369024\pi$
$80$	126567.			2.21104
$81$		0			0
$82$	−14692.7			−0.241305
$83$	100089.			1.59474			0.797369	−	0.603492i	$-0.206225\pi$
$83$	100089.			1.59474			0.797369	+	0.603492i	$0.206225\pi$
$84$		0			0
$85$	45803.5			0.687625
$86$	8878.17			0.129443
$87$		0			0
$88$		−	17331.8i		−	0.238581i
$89$	−65919.5			−0.882143			−0.441071	−	0.897472i	$-0.645401\pi$
$89$	−65919.5			−0.882143			−0.441071	+	0.897472i	$0.645401\pi$
$90$		0			0
$91$		−	24982.0i		−	0.316245i
$92$	14111.3	−	9835.72i	0.173820	−	0.121154i
$93$		0			0
$94$	75277.7			0.878712
$95$			53318.6i			0.606135i
$96$		0			0
$97$			127964.i			1.38089i	0.723384	+	0.690446i	$0.242586\pi$
$97$			127964.i			1.38089i	−0.723384	+	0.690446i	$0.757414\pi$
$98$			57930.0i			0.609310i
$99$		0			0
$100$	−54870.5			−0.548705
$101$			100006.i			0.975490i	0.872986	+	0.487745i	$0.162180\pi$
$101$			100006.i			0.975490i	−0.872986	+	0.487745i	$0.837820\pi$
$102$		0			0
$103$			98601.5i			0.915779i	0.889009	+	0.457889i	$0.151394\pi$
$103$			98601.5i			0.915779i	−0.889009	+	0.457889i	$0.848606\pi$
$104$		−	45291.4i		−	0.410612i
$105$		0			0
$106$			99698.3i			0.861833i
$107$	−41792.4			−0.352889			−0.176445	−	0.984311i	$-0.556460\pi$
$107$	−41792.4			−0.352889			−0.176445	+	0.984311i	$0.556460\pi$
$108$		0			0
$109$		−	233364.i		−	1.88134i	−0.339316	−	0.940672i	$-0.610196\pi$
$109$		−	233364.i		−	1.88134i	0.339316	−	0.940672i	$-0.389804\pi$
$110$			72787.3i			0.573553i
$111$		0			0
$112$		−	103520.i		−	0.779796i
$113$	−171057.			−1.26022			−0.630109	−	0.776507i	$-0.716990\pi$
$113$	−171057.			−1.26022			−0.630109	+	0.776507i	$0.716990\pi$
$114$		0			0
$115$	220441.	−	153650.i	1.55435	−	1.08339i
$116$			23041.6i			0.158989i
$117$		0			0
$118$	−187216.			−1.23776
$119$		−	37463.0i		−	0.242513i
$120$		0			0
$121$	−148873.			−0.924382
$122$	85697.5			0.521277
$123$		0			0
$124$	−48793.6			−0.284976
$125$	−526179.			−3.01203
$126$		0			0
$127$	−228241.			−1.25570			−0.627848	−	0.778336i	$-0.716064\pi$
$127$	−228241.			−1.25570			−0.627848	+	0.778336i	$0.716064\pi$
$128$		−	221753.i		−	1.19631i
$129$		0			0
$130$			190207.i			0.987118i
$131$		−	344449.i		−	1.75366i	−0.480798	−	0.876831i	$-0.659653\pi$
$131$		−	344449.i		−	1.75366i	0.480798	−	0.876831i	$-0.340347\pi$
$132$		0			0
$133$	43609.7			0.213773
$134$	124310.			0.598060
$135$		0			0
$136$		−	67918.9i		−	0.314879i
$137$	−337809.			−1.53769			−0.768847	−	0.639433i	$-0.779169\pi$
$137$	−337809.			−1.53769			−0.768847	+	0.639433i	$0.779169\pi$
$138$		0			0
$139$	−87339.6			−0.383420			−0.191710	−	0.981452i	$-0.561403\pi$
$139$	−87339.6			−0.383420			−0.191710	+	0.981452i	$0.561403\pi$
$140$			62208.5i			0.268244i
$141$		0			0
$142$	368651.			1.53425
$143$	31824.5			0.130143
$144$		0			0
$145$			359946.i			1.42173i
$146$		−	266819.i		−	1.03594i
$147$		0			0
$148$			42816.1i			0.160677i
$149$	490566.			1.81022			0.905111	−	0.425174i	$-0.139787\pi$
$149$	490566.			1.81022			0.905111	+	0.425174i	$0.139787\pi$
$150$		0			0
$151$	164972.			0.588799			0.294400	−	0.955682i	$-0.404880\pi$
$151$	164972.			0.588799			0.294400	+	0.955682i	$0.404880\pi$
$152$	79062.5			0.277563
$153$		0			0
$154$	59533.3			0.202282
$155$	−762233.			−2.54835
$156$		0			0
$157$			23450.8i			0.0759291i	0.999279	+	0.0379645i	$0.0120874\pi$
$157$			23450.8i			0.0759291i	−0.999279	+	0.0379645i	$0.987913\pi$
$158$	79916.0			0.254678
$159$		0			0
$160$			255882.i			0.790206i
$161$	−125671.	−	180301.i	−0.382095	−	0.548192i
$162$		0			0
$163$	−333766.			−0.983951			−0.491975	−	0.870609i	$-0.663725\pi$
$163$	−333766.			−0.983951			−0.491975	+	0.870609i	$0.663725\pi$
$164$		−	15996.7i		−	0.0464430i
$165$		0			0
$166$			623288.i			1.75557i
$167$		−	551527.i		−	1.53030i	−0.643855	−	0.765148i	$-0.722666\pi$
$167$		−	551527.i		−	1.53030i	0.643855	−	0.765148i	$-0.277334\pi$
$168$		0			0
$169$	−288129.			−0.776016
$170$			285235.i			0.756973i
$171$		0			0
$172$			9666.09i			0.0249132i
$173$			192759.i			0.489667i	0.969565	+	0.244833i	$0.0787332\pi$
$173$			192759.i			0.489667i	−0.969565	+	0.244833i	$0.921267\pi$
$174$		0			0
$175$			701080.i			1.73050i
$176$	131874.			0.320906
$177$		0			0
$178$		−	410505.i		−	0.971109i
$179$		−	708830.i		−	1.65352i	−0.562554	−	0.826761i	$-0.690181\pi$
$179$		−	708830.i		−	1.65352i	0.562554	−	0.826761i	$-0.309819\pi$
$180$		0			0
$181$		−	641358.i		−	1.45514i	−0.686035	−	0.727568i	$-0.740650\pi$
$181$		−	641358.i		−	1.45514i	0.686035	−	0.727568i	$-0.259350\pi$
$182$	155572.			0.348140
$183$		0			0
$184$	−227836.	−	326877.i	−0.496111	−	0.711771i
$185$			668855.i			1.43682i
$186$		0			0
$187$	47724.0			0.0998005
$188$			81958.5i			0.169122i
$189$		0			0
$190$	−332034.			−0.667265
$191$	365608.			0.725156			0.362578	−	0.931953i	$-0.381897\pi$
$191$	365608.			0.725156			0.362578	+	0.931953i	$0.381897\pi$
$192$		0			0
$193$	−178883.			−0.345680			−0.172840	−	0.984950i	$-0.555294\pi$
$193$	−178883.			−0.345680			−0.172840	+	0.984950i	$0.555294\pi$
$194$	−796880.			−1.52016
$195$		0			0
$196$	−63071.2			−0.117271
$197$			448036.i			0.822521i	0.911518	+	0.411261i	$0.134911\pi$
$197$			448036.i			0.822521i	−0.911518	+	0.411261i	$0.865089\pi$
$198$		0			0
$199$			838464.i			1.50090i	0.660928	+	0.750450i	$0.270163\pi$
$199$			838464.i			1.50090i	−0.660928	+	0.750450i	$0.729837\pi$
$200$			1.27103e6i			2.24688i
$201$		0			0
$202$	−622774.			−1.07387
$203$	294403.			0.501421
$204$		0			0
$205$		−	249893.i		−	0.415307i
$206$	−614027.			−1.00814
$207$		0			0
$208$	344613.			0.552298
$209$			55554.1i			0.0879732i
$210$		0			0
$211$	98823.7			0.152811			0.0764056	−	0.997077i	$-0.475656\pi$
$211$	98823.7			0.152811			0.0764056	+	0.997077i	$0.475656\pi$
$212$	−108546.			−0.165873
$213$		0			0
$214$		−	260257.i		−	0.388479i
$215$			151000.i			0.222782i
$216$		0			0
$217$			623436.i			0.898757i
$218$	1.45324e6			2.07108
$219$		0			0
$220$	−79247.0			−0.110389
$221$	124712.			0.171762
$222$		0			0
$223$	64321.7			0.0866154			0.0433077	−	0.999062i	$-0.486210\pi$
$223$	64321.7			0.0866154			0.0433077	+	0.999062i	$0.486210\pi$
$224$	209288.			0.278692
$225$		0			0
$226$		−	1.06524e6i		−	1.38731i
$227$	597491.			0.769603			0.384801	−	0.922999i	$-0.374270\pi$
$227$	597491.			0.769603			0.384801	+	0.922999i	$0.374270\pi$
$228$		0			0
$229$			992228.i			1.25033i	0.780495	+	0.625163i	$0.214967\pi$
$229$			992228.i			1.25033i	−0.780495	+	0.625163i	$0.785033\pi$
$230$	956831.	+	1.37277e6i	1.19266	+	1.71111i
$231$		0			0
$232$	533740.			0.651043
$233$		−	1.24078e6i		−	1.49729i	−0.662972	−	0.748644i	$-0.730705\pi$
$233$		−	1.24078e6i		−	1.49729i	0.662972	−	0.748644i	$-0.269295\pi$
$234$		0			0
$235$			1.28032e6i			1.51234i
$236$		−	203831.i		−	0.238227i
$237$		0			0
$238$	233296.			0.266971
$239$		−	1.50821e6i		−	1.70792i	−0.520341	−	0.853959i	$-0.674195\pi$
$239$		−	1.50821e6i		−	1.70792i	0.520341	−	0.853959i	$-0.325805\pi$
$240$		0			0
$241$		−	569092.i		−	0.631161i	−0.948899	−	0.315580i	$-0.897801\pi$
$241$		−	569092.i		−	0.631161i	0.948899	−	0.315580i	$-0.102199\pi$
$242$		−	927084.i		−	1.01761i
$243$		0			0
$244$			93303.0i			0.100328i
$245$	−985272.			−1.04868
$246$		0			0
$247$			145174.i			0.151407i
$248$			1.13026e6i			1.16694i
$249$		0			0
$250$		−	3.27671e6i		−	3.31580i
$251$	1.41542e6			1.41808			0.709038	−	0.705170i	$-0.249129\pi$
$251$	1.41542e6			1.41808			0.709038	+	0.705170i	$0.249129\pi$
$252$		0			0
$253$	229684.	−	160092.i	0.225595	−	0.157242i
$254$		−	1.42134e6i		−	1.38234i
$255$		0			0
$256$	638697.			0.609109
$257$			1.13148e6i			1.06860i	0.845296	+	0.534298i	$0.179424\pi$
$257$			1.13148e6i			1.06860i	−0.845296	+	0.534298i	$0.820576\pi$
$258$		0			0
$259$	547061.			0.506742
$260$	−207088.			−0.189986
$261$		0			0
$262$	2.14501e6			1.93052
$263$	318686.			0.284102			0.142051	−	0.989859i	$-0.454630\pi$
$263$	318686.			0.284102			0.142051	+	0.989859i	$0.454630\pi$
$264$		0			0
$265$	−1.69567e6			−1.48329
$266$			271573.i			0.235333i
$267$		0			0
$268$			135343.i			0.115106i
$269$			860814.i			0.725319i	0.931922	+	0.362659i	$0.118131\pi$
$269$			860814.i			0.725319i	−0.931922	+	0.362659i	$0.881869\pi$
$270$		0			0
$271$	1.82091e6			1.50614			0.753069	−	0.657941i	$-0.228572\pi$
$271$	1.82091e6			1.50614			0.753069	+	0.657941i	$0.228572\pi$
$272$	516782.			0.423531
$273$		0			0
$274$		−	2.10366e6i		−	1.69277i
$275$	−893103.			−0.712147
$276$		0			0
$277$	−178219.			−0.139558			−0.0697790	−	0.997562i	$-0.522229\pi$
$277$	−178219.			−0.139558			−0.0697790	+	0.997562i	$0.522229\pi$
$278$		−	543895.i		−	0.422088i
$279$		0			0
$280$	1.44101e6			1.09843
$281$	−2.40132e6			−1.81419			−0.907096	−	0.420923i	$-0.861706\pi$
$281$	−2.40132e6			−1.81419			−0.907096	+	0.420923i	$0.861706\pi$
$282$		0			0
$283$		−	185582.i		−	0.137743i	−0.997626	−	0.0688716i	$-0.978060\pi$
$283$		−	185582.i		−	0.137743i	0.997626	−	0.0688716i	$-0.0219399\pi$
$284$			401369.i			0.295289i
$285$		0			0
$286$			198183.i			0.143268i
$287$	−204390.			−0.146472
$288$		0			0
$289$	−1.23284e6			−0.868284
$290$	−2.24152e6			−1.56512
$291$		0			0
$292$	290499.			0.199382
$293$	602792.			0.410202			0.205101	−	0.978741i	$-0.434248\pi$
$293$	602792.			0.410202			0.205101	+	0.978741i	$0.434248\pi$
$294$		0			0
$295$		−	3.18416e6i		−	2.13030i
$296$	991799.			0.657952
$297$		0			0
$298$			3.05493e6i			1.99279i
$299$	600210.	−	418351.i	0.388262	−	0.270622i
$300$		0			0
$301$	123504.			0.0785713
$302$			1.02734e6i			0.648181i
$303$		0			0
$304$			601571.i			0.373339i
$305$			1.45754e6i			0.897162i
$306$		0			0
$307$	−845284.			−0.511866			−0.255933	−	0.966694i	$-0.582383\pi$
$307$	−845284.			−0.511866			−0.255933	+	0.966694i	$0.582383\pi$
$308$			64816.8i			0.0389323i
$309$		0			0
$310$		−	4.74670e6i		−	2.80535i
$311$			1.60688e6i			0.942070i	0.882114	+	0.471035i	$0.156120\pi$
$311$			1.60688e6i			0.942070i	−0.882114	+	0.471035i	$0.843880\pi$
$312$		0			0
$313$		−	2.54513e6i		−	1.46842i	−0.678924	−	0.734208i	$-0.737553\pi$
$313$		−	2.54513e6i		−	1.46842i	0.678924	−	0.734208i	$-0.262447\pi$
$314$	−146037.			−0.0835867
$315$		0			0
$316$			87008.4i			0.0490166i
$317$		−	1.20894e6i		−	0.675702i	−0.941200	−	0.337851i	$-0.890300\pi$
$317$		−	1.20894e6i		−	0.675702i	0.941200	−	0.337851i	$-0.109700\pi$
$318$		0			0
$319$			375038.i			0.206347i
$320$	2.45668e6			1.34114
$321$		0			0
$322$	1.12280e6	−	782600.i	0.603478	−	0.420630i
$323$			217703.i			0.116107i
$324$		0			0
$325$	−2.33385e6			−1.22565
$326$		−	2.07848e6i		−	1.08318i
$327$		0			0
$328$	−370550.			−0.190179
$329$	1.04718e6			0.533376
$330$		0			0
$331$	−2.81512e6			−1.41230			−0.706151	−	0.708062i	$-0.749570\pi$
$331$	−2.81512e6			−1.41230			−0.706151	+	0.708062i	$0.749570\pi$
$332$	−678604.			−0.337887
$333$		0			0
$334$	3.43456e6			1.68463
$335$			2.11426e6i			1.02931i
$336$		0			0
$337$			1.94718e6i			0.933966i	0.884266	+	0.466983i	$0.154659\pi$
$337$			1.94718e6i			0.933966i	−0.884266	+	0.466983i	$0.845341\pi$
$338$		−	1.79429e6i		−	0.854279i
$339$		0			0
$340$	−310549.			−0.145691
$341$	−794192.			−0.369862
$342$		0			0
$343$			2.26183e6i			1.03806i
$344$	223907.			0.102017
$345$		0			0
$346$	−1.20038e6			−0.539051
$347$		−	2.50387e6i		−	1.11632i	−0.829735	−	0.558158i	$-0.811508\pi$
$347$		−	2.50387e6i		−	1.11632i	0.829735	−	0.558158i	$-0.188492\pi$
$348$		0			0
$349$	863267.			0.379386			0.189693	−	0.981843i	$-0.439251\pi$
$349$	863267.			0.379386			0.189693	+	0.981843i	$0.439251\pi$
$350$	−4.36588e6			−1.90503
$351$		0			0
$352$			266611.i			0.114689i
$353$		−	642585.i		−	0.274469i	−0.990539	−	0.137235i	$-0.956179\pi$
$353$		−	642585.i		−	0.274469i	0.990539	−	0.137235i	$-0.0438215\pi$
$354$		0			0
$355$			6.27001e6i			2.64057i
$356$	446936.			0.186905
$357$		0			0
$358$	4.41414e6			1.82028
$359$	−3.70922e6			−1.51896			−0.759480	−	0.650531i	$-0.774546\pi$
$359$	−3.70922e6			−1.51896			−0.759480	+	0.650531i	$0.774546\pi$
$360$		0			0
$361$	2.22268e6			0.897653
$362$	3.99397e6			1.60189
$363$		0			0
$364$			169379.i			0.0670048i
$365$	4.53805e6			1.78294
$366$		0			0
$367$			3.18123e6i			1.23291i	0.787392	+	0.616453i	$0.211431\pi$
$367$			3.18123e6i			1.23291i	−0.787392	+	0.616453i	$0.788569\pi$
$368$	2.48715e6	−	1.73356e6i	0.957375	−	0.667298i
$369$		0			0
$370$	−4.16520e6			−1.58173
$371$			1.38690e6i			0.523131i
$372$		0			0
$373$			2.68936e6i			1.00087i	0.865774	+	0.500435i	$0.166826\pi$
$373$			2.68936e6i			1.00087i	−0.865774	+	0.500435i	$0.833174\pi$
$374$			297194.i			0.109866i
$375$		0			0
$376$	1.89850e6			0.692534
$377$			980049.i			0.355136i
$378$		0			0
$379$		−	1.79803e6i		−	0.642984i	−0.946912	−	0.321492i	$-0.895816\pi$
$379$		−	1.79803e6i		−	0.642984i	0.946912	−	0.321492i	$-0.104184\pi$
$380$		−	361502.i		−	0.128426i
$381$		0			0
$382$			2.27677e6i			0.798290i
$383$	−2.62741e6			−0.915231			−0.457616	−	0.889150i	$-0.651296\pi$
$383$	−2.62741e6			−0.915231			−0.457616	+	0.889150i	$0.651296\pi$
$384$		0			0
$385$			1.01254e6i			0.348145i
$386$		−	1.11397e6i		−	0.380543i
$387$		0			0
$388$		−	867603.i		−	0.292578i
$389$	1.87693e6			0.628890			0.314445	−	0.949276i	$-0.398182\pi$
$389$	1.87693e6			0.628890			0.314445	+	0.949276i	$0.398182\pi$
$390$		0			0
$391$	900074.	−	627360.i	0.297740	−	0.207527i
$392$			1.46099e6i			0.480212i
$393$		0			0
$394$	−2.79008e6			−0.905474
$395$			1.35921e6i			0.438322i
$396$		0			0
$397$	−2.51880e6			−0.802079			−0.401040	−	0.916061i	$-0.631351\pi$
$397$	−2.51880e6			−0.802079			−0.401040	+	0.916061i	$0.631351\pi$
$398$	−5.22142e6			−1.65227
$399$		0			0
$400$	−9.67101e6			−3.02219
$401$	5.93344e6			1.84266			0.921331	−	0.388780i	$-0.127103\pi$
$401$	5.93344e6			1.84266			0.921331	+	0.388780i	$0.127103\pi$
$402$		0			0
$403$	−2.07538e6			−0.636554
$404$		−	678044.i		−	0.206683i
$405$		0			0
$406$			1.83335e6i			0.551990i
$407$			696899.i			0.208537i
$408$		0			0
$409$	2.11520e6			0.625236			0.312618	−	0.949879i	$-0.398794\pi$
$409$	2.11520e6			0.625236			0.312618	+	0.949879i	$0.398794\pi$
$410$	1.55618e6			0.457192
$411$		0			0
$412$		−	668522.i		−	0.194032i
$413$	−2.60435e6			−0.751320
$414$		0			0
$415$	−1.06009e7			−3.02149
$416$			696707.i			0.197386i
$417$		0			0
$418$	−345956.			−0.0968455
$419$	−2.88006e6			−0.801430			−0.400715	−	0.916203i	$-0.631238\pi$
$419$	−2.88006e6			−0.801430			−0.400715	+	0.916203i	$0.631238\pi$
$420$		0			0
$421$			2.94775e6i			0.810561i	0.914192	+	0.405281i	$0.132826\pi$
$421$			2.94775e6i			0.810561i	−0.914192	+	0.405281i	$0.867174\pi$
$422$			615411.i			0.168223i
$423$		0			0
$424$			2.51439e6i			0.679231i
$425$	−3.49985e6			−0.939889
$426$		0			0
$427$	1.19213e6			0.316414
$428$	283354.			0.0747687
$429$		0			0
$430$	−940329.			−0.245250
$431$	309698.			0.0803056			0.0401528	−	0.999194i	$-0.487216\pi$
$431$	309698.			0.0803056			0.0401528	+	0.999194i	$0.487216\pi$
$432$		0			0
$433$		−	4.92862e6i		−	1.26330i	−0.775254	−	0.631649i	$-0.782378\pi$
$433$		−	4.92862e6i		−	1.26330i	0.775254	−	0.631649i	$-0.217622\pi$
$434$	−3.88236e6			−0.989399
$435$		0			0
$436$			1.58222e6i			0.398612i
$437$	730291.	+	1.04775e6i	0.182933	+	0.262455i
$438$		0			0
$439$	2.58300e6			0.639681			0.319841	−	0.947471i	$-0.396371\pi$
$439$	2.58300e6			0.639681			0.319841	+	0.947471i	$0.396371\pi$
$440$			1.83569e6i			0.452031i
$441$		0			0
$442$			776628.i			0.189085i
$443$		−	1.11272e6i		−	0.269386i	−0.990887	−	0.134693i	$-0.956995\pi$
$443$		−	1.11272e6i		−	0.269386i	0.990887	−	0.134693i	$-0.0430048\pi$
$444$		0			0
$445$	6.98185e6			1.67136
$446$			400554.i			0.0953508i
$447$		0			0
$448$		−	2.00934e6i		−	0.472997i
$449$		−	3.46988e6i		−	0.812266i	−0.913814	−	0.406133i	$-0.866877\pi$
$449$		−	3.46988e6i		−	0.812266i	0.913814	−	0.406133i	$-0.133123\pi$
$450$		0			0
$451$		−	260371.i		−	0.0602769i
$452$	1.15977e6			0.267010
$453$		0			0
$454$			3.72079e6i			0.847219i
$455$			2.64597e6i			0.599178i
$456$		0			0
$457$			959210.i			0.214844i	0.994214	+	0.107422i	$0.0342596\pi$
$457$			959210.i			0.214844i	−0.994214	+	0.107422i	$0.965740\pi$
$458$	−6.17897e6			−1.37642
$459$		0			0
$460$	−1.49460e6	+	1.04175e6i	−0.329329	+	0.229545i
$461$		−	8.49982e6i		−	1.86276i	−0.364047	−	0.931381i	$-0.618605\pi$
$461$		−	8.49982e6i		−	1.86276i	0.364047	−	0.931381i	$-0.381395\pi$
$462$		0			0
$463$	156163.			0.0338552			0.0169276	−	0.999857i	$-0.494612\pi$
$463$	156163.			0.0338552			0.0169276	+	0.999857i	$0.494612\pi$
$464$			4.06112e6i			0.875692i
$465$		0			0
$466$	7.72679e6			1.64829
$467$	−2.58234e6			−0.547925			−0.273962	−	0.961740i	$-0.588334\pi$
$467$	−2.58234e6			−0.547925			−0.273962	+	0.961740i	$0.588334\pi$
$468$		0			0
$469$	1.72927e6			0.363021
$470$	−7.97302e6			−1.66486
$471$		0			0
$472$	−4.72158e6			−0.975511
$473$			157331.i			0.0323341i
$474$		0			0
$475$		−	4.07407e6i		−	0.828504i
$476$			254001.i			0.0513828i
$477$		0			0
$478$	9.39216e6			1.88016
$479$	371115.			0.0739043			0.0369521	−	0.999317i	$-0.488235\pi$
$479$	371115.			0.0739043			0.0369521	+	0.999317i	$0.488235\pi$
$480$		0			0
$481$			1.82113e6i			0.358905i
$482$	3.54394e6			0.694815
$483$		0			0
$484$	1.00936e6			0.195854
$485$		−	1.35533e7i		−	2.61632i
$486$		0			0
$487$	3.54926e6			0.678135			0.339067	−	0.940762i	$-0.389889\pi$
$487$	3.54926e6			0.678135			0.339067	+	0.940762i	$0.389889\pi$
$488$	2.16129e6			0.410831
$489$		0			0
$490$		−	6.13565e6i		−	1.15444i
$491$			3.41804e6i			0.639843i	0.947444	+	0.319921i	$0.103657\pi$
$491$			3.41804e6i			0.639843i	−0.947444	+	0.319921i	$0.896343\pi$
$492$		0			0
$493$			1.46968e6i			0.272337i
$494$	−904050.			−0.166677
$495$		0			0
$496$	−8.59995e6			−1.56961
$497$	5.12829e6			0.931283
$498$		0			0
$499$	2.46036e6			0.442331			0.221166	−	0.975236i	$-0.429014\pi$
$499$	2.46036e6			0.442331			0.221166	+	0.975236i	$0.429014\pi$
$500$	3.56751e6			0.638176
$501$		0			0
$502$			8.81430e6i			1.56109i
$503$	3.93938e6			0.694238			0.347119	−	0.937821i	$-0.387160\pi$
$503$	3.93938e6			0.694238			0.347119	+	0.937821i	$0.387160\pi$
$504$		0			0
$505$		−	1.05921e7i		−	1.84822i
$506$	996949.	+	1.43033e6i	0.173100	+	0.248347i
$507$		0			0
$508$	1.54748e6			0.266052
$509$		−	6.17050e6i		−	1.05566i	−0.849349	−	0.527832i	$-0.823005\pi$
$509$		−	6.17050e6i		−	1.05566i	0.849349	−	0.527832i	$-0.176995\pi$
$510$		0			0
$511$		−	3.71170e6i		−	0.628812i
$512$		−	3.11869e6i		−	0.525771i
$513$		0			0
$514$	−7.04613e6			−1.17637
$515$		−	1.04434e7i		−	1.73509i
$516$		0			0
$517$			1.33400e6i			0.219498i
$518$			3.40675e6i			0.557848i
$519$		0			0
$520$			4.79703e6i			0.777972i
$521$	−4.40855e6			−0.711543			−0.355772	−	0.934573i	$-0.615782\pi$
$521$	−4.40855e6			−0.711543			−0.355772	+	0.934573i	$0.615782\pi$
$522$		0			0
$523$		−	8.64117e6i		−	1.38140i	−0.723143	−	0.690699i	$-0.757303\pi$
$523$		−	8.64117e6i		−	1.38140i	0.723143	−	0.690699i	$-0.242697\pi$
$524$			2.33537e6i			0.371559i
$525$		0			0
$526$			1.98458e6i			0.312754i
$527$	−3.11224e6			−0.488142
$528$		0			0
$529$	2.22734e6	−	6.03866e6i	0.346057	−	0.938213i
$530$		−	1.05595e7i		−	1.63288i
$531$		0			0
$532$	−295675.			−0.0452935
$533$		−	680400.i		−	0.103740i
$534$		0			0
$535$	4.42644e6			0.668605
$536$	3.13510e6			0.471345
$537$		0			0
$538$	−5.36060e6			−0.798469
$539$	−1.02658e6			−0.152203
$540$		0			0
$541$	7.51518e6			1.10394			0.551971	−	0.833863i	$-0.313876\pi$
$541$	7.51518e6			1.10394			0.551971	+	0.833863i	$0.313876\pi$
$542$			1.13395e7i			1.65804i
$543$		0			0
$544$			1.04478e6i			0.151366i
$545$			2.47167e7i			3.56451i
$546$		0			0
$547$	−5.55411e6			−0.793681			−0.396841	−	0.917888i	$-0.629893\pi$
$547$	−5.55411e6			−0.793681			−0.396841	+	0.917888i	$0.629893\pi$
$548$	2.29036e6			0.325800
$549$		0			0
$550$		−	5.56167e6i		−	0.783969i
$551$	−1.71081e6			−0.240062
$552$		0			0
$553$	1.11171e6			0.154589
$554$		−	1.10983e6i		−	0.153633i
$555$		0			0
$556$	592166.			0.0812374
$557$	1.32542e7			1.81015			0.905076	−	0.425250i	$-0.139814\pi$
$557$	1.32542e7			1.81015			0.905076	+	0.425250i	$0.139814\pi$
$558$		0			0
$559$			411136.i			0.0556489i
$560$			1.09643e7i			1.47745i
$561$		0			0
$562$		−	1.49539e7i		−	1.99716i
$563$	560797.			0.0745650			0.0372825	−	0.999305i	$-0.488130\pi$
$563$	560797.			0.0745650			0.0372825	+	0.999305i	$0.488130\pi$
$564$		0			0
$565$	1.81175e7			2.38769
$566$	1.15569e6			0.151635
$567$		0			0
$568$	9.29737e6			1.20918
$569$	−1.10699e7			−1.43338			−0.716690	−	0.697392i	$-0.754344\pi$
$569$	−1.10699e7			−1.43338			−0.716690	+	0.697392i	$0.754344\pi$
$570$		0			0
$571$			1.13644e6i			0.145867i	0.997337	+	0.0729333i	$0.0232360\pi$
$571$			1.13644e6i			0.145867i	−0.997337	+	0.0729333i	$0.976764\pi$
$572$	−215771.			−0.0275742
$573$		0			0
$574$		−	1.27281e6i		−	0.161244i
$575$	−1.68439e7	+	1.17404e7i	−2.12458	+	1.48085i
$576$		0			0
$577$	−1.25931e7			−1.57468			−0.787340	−	0.616519i	$-0.788543\pi$
$577$	−1.25931e7			−1.57468			−0.787340	+	0.616519i	$0.788543\pi$
$578$		−	7.67733e6i		−	0.955852i
$579$		0			0
$580$		−	2.44045e6i		−	0.301231i
$581$			8.67053e6i			1.06563i
$582$		0			0
$583$	−1.76676e6			−0.215282
$584$		−	6.72916e6i		−	0.816448i
$585$		0			0
$586$			3.75380e6i			0.451572i
$587$			8.04591e6i			0.963785i	0.876230	+	0.481892i	$0.160050\pi$
$587$			8.04591e6i			0.963785i	−0.876230	+	0.481892i	$0.839950\pi$
$588$		0			0
$589$		−	3.62287e6i		−	0.430293i
$590$	1.98289e7			2.34514
$591$		0			0
$592$			7.54641e6i			0.884985i
$593$			4.26404e6i			0.497949i	0.968510	+	0.248975i	$0.0800936\pi$
$593$			4.26404e6i			0.497949i	−0.968510	+	0.248975i	$0.919906\pi$
$594$		0			0
$595$			3.96789e6i			0.459481i
$596$	−3.32606e6			−0.383543
$597$		0			0
$598$	2.60523e6	+	3.73772e6i	0.297915	+	0.427419i
$599$		−	4.17933e6i		−	0.475926i	−0.971274	−	0.237963i	$-0.923520\pi$
$599$		−	4.17933e6i		−	0.475926i	0.971274	−	0.237963i	$-0.0764797\pi$
$600$		0			0
$601$	−5.17862e6			−0.584828			−0.292414	−	0.956292i	$-0.594458\pi$
$601$	−5.17862e6			−0.584828			−0.292414	+	0.956292i	$0.594458\pi$
$602$			769103.i			0.0864954i
$603$		0			0
$604$	−1.11851e6			−0.124752
$605$	1.57678e7			1.75139
$606$		0			0
$607$	−9.30661e6			−1.02523			−0.512613	−	0.858620i	$-0.671322\pi$
$607$	−9.30661e6			−1.02523			−0.512613	+	0.858620i	$0.671322\pi$
$608$	−1.21620e6			−0.133428
$609$		0			0
$610$	−9.07663e6			−0.987643
$611$			3.48601e6i			0.377769i
$612$		0			0
$613$		−	1.12511e7i		−	1.20932i	−0.796483	−	0.604661i	$-0.793309\pi$
$613$		−	1.12511e7i		−	1.20932i	0.796483	−	0.604661i	$-0.206691\pi$
$614$		−	5.26389e6i		−	0.563489i
$615$		0			0
$616$	1.50143e6			0.159424
$617$	1.09319e7			1.15606			0.578031	−	0.816015i	$-0.303821\pi$
$617$	1.09319e7			1.15606			0.578031	+	0.816015i	$0.303821\pi$
$618$		0			0
$619$		−	2.72353e6i		−	0.285697i	−0.989745	−	0.142849i	$-0.954374\pi$
$619$		−	2.72353e6i		−	0.285697i	0.989745	−	0.142849i	$-0.0456262\pi$
$620$	5.16796e6			0.539933
$621$		0			0
$622$	−1.00066e7			−1.03708
$623$		−	5.71051e6i		−	0.589461i
$624$		0			0
$625$	3.04397e7			3.11703
$626$	1.58495e7			1.61651
$627$		0			0
$628$		−	158997.i		−	0.0160876i
$629$			2.73097e6i			0.275227i
$630$		0			0
$631$		−	1.54067e7i		−	1.54041i	−0.637799	−	0.770203i	$-0.720155\pi$
$631$		−	1.54067e7i		−	1.54041i	0.637799	−	0.770203i	$-0.279845\pi$
$632$	2.01548e6			0.200718
$633$		0			0
$634$	7.52849e6			0.743848
$635$	2.41741e7			2.37912
$636$		0			0
$637$	−2.68266e6			−0.261950
$638$	−2.33550e6			−0.227158
$639$		0			0
$640$			2.34869e7i			2.26660i
$641$	1.16178e6			0.111681			0.0558406	−	0.998440i	$-0.482216\pi$
$641$	1.16178e6			0.111681			0.0558406	+	0.998440i	$0.482216\pi$
$642$		0			0
$643$		−	1.44536e7i		−	1.37864i	−0.724459	−	0.689318i	$-0.757910\pi$
$643$		−	1.44536e7i		−	1.37864i	0.724459	−	0.689318i	$-0.242090\pi$
$644$	852055.	+	1.22244e6i	0.0809567	+	0.116149i
$645$		0			0
$646$	−1.35571e6			−0.127816
$647$			7.40717e6i			0.695651i	0.937559	+	0.347825i	$0.113080\pi$
$647$			7.40717e6i			0.695651i	−0.937559	+	0.347825i	$0.886920\pi$
$648$		0			0
$649$		−	3.31767e6i		−	0.309187i
$650$		−	1.45337e7i		−	1.34926i
$651$		0			0
$652$	2.26295e6			0.208476
$653$			1.02302e7i			0.938859i	0.882970	+	0.469430i	$0.155540\pi$
$653$			1.02302e7i			0.938859i	−0.882970	+	0.469430i	$0.844460\pi$
$654$		0			0
$655$			3.64822e7i			3.32260i
$656$		−	2.81944e6i		−	0.255802i
$657$		0			0
$658$			6.52119e6i			0.587168i
$659$	−1.64316e7			−1.47389			−0.736946	−	0.675952i	$-0.763733\pi$
$659$	−1.64316e7			−1.47389			−0.736946	+	0.675952i	$0.763733\pi$
$660$		0			0
$661$		−	2.05635e7i		−	1.83060i	−0.402769	−	0.915302i	$-0.631952\pi$
$661$		−	2.05635e7i		−	1.83060i	0.402769	−	0.915302i	$-0.368048\pi$
$662$		−	1.75308e7i		−	1.55474i
$663$		0			0
$664$			1.57193e7i			1.38361i
$665$	−4.61891e6			−0.405028
$666$		0			0
$667$	4.93010e6	+	7.07323e6i	0.429083	+	0.615606i
$668$			3.73937e6i			0.324233i
$669$		0			0
$670$	−1.31663e7			−1.13312
$671$			1.51865e6i			0.130212i
$672$		0			0
$673$	−7.28031e6			−0.619602			−0.309801	−	0.950801i	$-0.600262\pi$
$673$	−7.28031e6			−0.619602			−0.309801	+	0.950801i	$0.600262\pi$
$674$	−1.21258e7			−1.02816
$675$		0			0
$676$	1.95353e6			0.164419
$677$	1.49285e7			1.25182			0.625912	−	0.779893i	$-0.284727\pi$
$677$	1.49285e7			1.25182			0.625912	+	0.779893i	$0.284727\pi$
$678$		0			0
$679$	−1.10854e7			−0.922732
$680$			7.19362e6i			0.596589i
$681$		0			0
$682$		−	4.94572e6i		−	0.407163i
$683$		−	1.18004e7i		−	0.967933i	−0.875087	−	0.483966i	$-0.839196\pi$
$683$		−	1.18004e7i		−	0.967933i	0.875087	−	0.483966i	$-0.160804\pi$
$684$		0			0
$685$	3.57790e7			2.91341
$686$	−1.40852e7			−1.14276
$687$		0			0
$688$			1.70367e6i			0.137219i
$689$	−4.61690e6			−0.370512
$690$		0			0
$691$	−2.09126e7			−1.66615			−0.833074	−	0.553162i	$-0.813421\pi$
$691$	−2.09126e7			−1.66615			−0.833074	+	0.553162i	$0.813421\pi$
$692$		−	1.30692e6i		−	0.103749i
$693$		0			0
$694$	1.55925e7			1.22890
$695$	9.25056e6			0.726451
$696$		0			0
$697$		−	1.02033e6i		−	0.0795532i
$698$			5.37588e6i			0.417648i
$699$		0			0
$700$		−	4.75335e6i		−	0.366652i
$701$	319263.			0.0245388			0.0122694	−	0.999925i	$-0.496094\pi$
$701$	319263.			0.0245388			0.0122694	+	0.999925i	$0.496094\pi$
$702$		0			0
$703$	−3.17905e6			−0.242610
$704$	2.55969e6			0.194651
$705$		0			0
$706$	4.00161e6			0.302150
$707$	−8.66338e6			−0.651837
$708$		0			0
$709$			1.69303e6i			0.126488i	0.997998	+	0.0632440i	$0.0201446\pi$
$709$			1.69303e6i			0.126488i	−0.997998	+	0.0632440i	$0.979855\pi$
$710$	−3.90456e7			−2.90688
$711$		0			0
$712$		−	1.03529e7i		−	0.765354i
$713$	−1.49785e7	+	1.04401e7i	−1.10343	+	0.769098i
$714$		0			0
$715$	−3.37068e6			−0.246577
$716$			4.80589e6i			0.350342i
$717$		0			0
$718$		−	2.30986e7i		−	1.67215i
$719$		−	2.12558e7i		−	1.53340i	−0.642007	−	0.766699i	$-0.721898\pi$
$719$		−	2.12558e7i		−	1.53340i	0.642007	−	0.766699i	$-0.278102\pi$
$720$		0			0
$721$	−8.54171e6			−0.611937
$722$			1.38414e7i			0.988183i
$723$		0			0
$724$			4.34843e6i			0.308309i
$725$		−	2.75035e7i		−	1.94331i
$726$		0			0
$727$		−	6.54119e6i		−	0.459009i	−0.973308	−	0.229504i	$-0.926289\pi$
$727$		−	6.54119e6i		−	0.459009i	0.973308	−	0.229504i	$-0.0737105\pi$
$728$	3.92352e6			0.274377
$729$		0			0
$730$			2.82601e7i			1.96275i
$731$			616540.i			0.0426744i
$732$		0			0
$733$		−	5.13617e6i		−	0.353085i	−0.984293	−	0.176542i	$-0.943509\pi$
$733$		−	5.13617e6i		−	0.353085i	0.984293	−	0.176542i	$-0.0564913\pi$
$734$	−1.98107e7			−1.35725
$735$		0			0
$736$	3.50476e6	+	5.02828e6i	0.238486	+	0.342157i
$737$			2.20291e6i			0.149392i
$738$		0			0
$739$	−1.82475e7			−1.22911			−0.614556	−	0.788873i	$-0.710665\pi$
$739$	−1.82475e7			−1.22911			−0.614556	+	0.788873i	$0.710665\pi$
$740$		−	4.53486e6i		−	0.304428i
$741$		0			0
$742$	−8.63672e6			−0.575889
$743$	7.23785e6			0.480992			0.240496	−	0.970650i	$-0.422690\pi$
$743$	7.23785e6			0.480992			0.240496	+	0.970650i	$0.422690\pi$
$744$		0			0
$745$	−5.19582e7			−3.42976
$746$	−1.67476e7			−1.10181
$747$		0			0
$748$	−323570.			−0.0211453
$749$		−	3.62042e6i		−	0.235806i
$750$		0			0
$751$			3.98771e6i			0.258003i	0.991644	+	0.129001i	$0.0411771\pi$
$751$			3.98771e6i			0.258003i	−0.991644	+	0.129001i	$0.958823\pi$
$752$			1.44453e7i			0.931499i
$753$		0			0
$754$	−6.10312e6			−0.390952
$755$	−1.74730e7			−1.11558
$756$		0			0
$757$			1.63486e7i			1.03691i	0.855106	+	0.518454i	$0.173492\pi$
$757$			1.63486e7i			1.03691i	−0.855106	+	0.518454i	$0.826508\pi$
$758$	1.11970e7			0.707830
$759$		0			0
$760$	−8.37389e6			−0.525888
$761$			1.57678e6i			0.0986981i	0.998782	+	0.0493491i	$0.0157147\pi$
$761$			1.57678e6i			0.0986981i	−0.998782	+	0.0493491i	$0.984285\pi$
$762$		0			0
$763$	2.02160e7			1.25714
$764$	−2.47883e6			−0.153643
$765$		0			0
$766$		−	1.63618e7i		−	1.00753i
$767$		−	8.66973e6i		−	0.532129i
$768$		0			0
$769$		−	1.64235e7i		−	1.00150i	−0.865592	−	0.500749i	$-0.833058\pi$
$769$		−	1.64235e7i		−	1.00150i	0.865592	−	0.500749i	$-0.166942\pi$
$770$	−6.30545e6			−0.383257
$771$		0			0
$772$	1.21283e6			0.0732414
$773$	−5.64862e6			−0.340012			−0.170006	−	0.985443i	$-0.554379\pi$
$773$	−5.64862e6			−0.340012			−0.170006	+	0.985443i	$0.554379\pi$
$774$		0			0
$775$	5.82422e7			3.48324
$776$	−2.00973e7			−1.19807
$777$		0			0
$778$			1.16883e7i			0.692315i
$779$	1.18773e6			0.0701255
$780$		0			0
$781$			6.53290e6i			0.383247i
$782$	3.90680e6	+	5.60509e6i	0.228457	+	0.327767i
$783$		0			0
$784$	−1.11164e7			−0.645913
$785$		−	2.48379e6i		−	0.143860i
$786$		0			0
$787$		−	413553.i		−	0.0238010i	−0.999929	−	0.0119005i	$-0.996212\pi$
$787$		−	413553.i		−	0.0238010i	0.999929	−	0.0119005i	$-0.00378813\pi$
$788$		−	3.03770e6i		−	0.174273i
$789$		0			0
$790$	−8.46428e6			−0.482528
$791$		−	1.48185e7i		−	0.842096i
$792$		0			0
$793$			3.96854e6i			0.224103i
$794$		−	1.56855e7i		−	0.882971i
$795$		0			0
$796$		−	5.68481e6i		−	0.318005i
$797$	9.25916e6			0.516328			0.258164	−	0.966101i	$-0.416882\pi$
$797$	9.25916e6			0.516328			0.258164	+	0.966101i	$0.416882\pi$
$798$		0			0
$799$			5.22762e6i			0.289692i
$800$		−	1.95520e7i		−	1.08010i
$801$		0			0
$802$			3.69497e7i			2.02850i
$803$	4.72832e6			0.258772
$804$		0			0
$805$	1.33104e7	+	1.90965e7i	0.723940	+	1.03864i
$806$		−	1.29241e7i		−	0.700752i
$807$		0			0
$808$	−1.57063e7			−0.846343
$809$			1.40211e7i			0.753201i	0.926376	+	0.376600i	$0.122907\pi$
$809$			1.40211e7i			0.753201i	−0.926376	+	0.376600i	$0.877093\pi$
$810$		0			0
$811$	8.48643e6			0.453078			0.226539	−	0.974002i	$-0.427259\pi$
$811$	8.48643e6			0.453078			0.226539	+	0.974002i	$0.427259\pi$
$812$	−1.99606e6			−0.106239
$813$		0			0
$814$	−4.33984e6			−0.229569
$815$	3.53508e7			1.86425
$816$		0			0
$817$	−717697.			−0.0376171
$818$			1.31721e7i			0.688292i
$819$		0			0
$820$			1.69428e6i			0.0879937i
$821$			1.56792e7i			0.811834i	0.913910	+	0.405917i	$0.133048\pi$
$821$			1.56792e7i			0.811834i	−0.913910	+	0.405917i	$0.866952\pi$
$822$		0			0
$823$	−1.40496e6			−0.0723041			−0.0361521	−	0.999346i	$-0.511510\pi$
$823$	−1.40496e6			−0.0723041			−0.0361521	+	0.999346i	$0.511510\pi$
$824$	−1.54857e7			−0.794537
$825$		0			0
$826$		−	1.62182e7i		−	0.827092i
$827$	1.63825e7			0.832947			0.416474	−	0.909148i	$-0.363266\pi$
$827$	1.63825e7			0.832947			0.416474	+	0.909148i	$0.363266\pi$
$828$		0			0
$829$	1.52878e7			0.772609			0.386305	−	0.922371i	$-0.373751\pi$
$829$	1.52878e7			0.772609			0.386305	+	0.922371i	$0.373751\pi$
$830$		−	6.60154e7i		−	3.32621i
$831$		0			0
$832$	6.68897e6			0.335005
$833$	−4.02292e6			−0.200876
$834$		0			0
$835$			5.84148e7i			2.89939i
$836$		−	376659.i		−	0.0186394i
$837$		0			0
$838$		−	1.79351e7i		−	0.882257i
$839$	−2.49913e7			−1.22570			−0.612850	−	0.790199i	$-0.709977\pi$
$839$	−2.49913e7			−1.22570			−0.612850	+	0.790199i	$0.709977\pi$
$840$		0			0
$841$	8.96167e6			0.436917
$842$	−1.83567e7			−0.892309
$843$		0			0
$844$	−670028.			−0.0323770
$845$	3.05172e7			1.47029
$846$		0			0
$847$		−	1.28966e7i		−	0.617685i
$848$	−1.91315e7			−0.913607
$849$		0			0
$850$		−	2.17948e7i		−	1.03468i
$851$	9.16114e6	+	1.31435e7i	0.433636	+	0.622139i
$852$		0			0
$853$	−1.09953e7			−0.517409			−0.258704	−	0.965957i	$-0.583296\pi$
$853$	−1.09953e7			−0.517409			−0.258704	+	0.965957i	$0.583296\pi$
$854$			7.42385e6i			0.348325i
$855$		0			0
$856$		−	6.56366e6i		−	0.306169i
$857$			2.24179e7i			1.04266i	0.853355	+	0.521330i	$0.174564\pi$
$857$			2.24179e7i			1.04266i	−0.853355	+	0.521330i	$0.825436\pi$
$858$		0			0
$859$	−2.68195e7			−1.24013			−0.620066	−	0.784550i	$-0.712894\pi$
$859$	−2.68195e7			−1.24013			−0.620066	+	0.784550i	$0.712894\pi$
$860$		−	1.02378e6i		−	0.0472021i
$861$		0			0
$862$			1.92860e6i			0.0884046i
$863$		−	1.44770e7i		−	0.661684i	−0.943686	−	0.330842i	$-0.892667\pi$
$863$		−	1.44770e7i		−	0.661684i	0.943686	−	0.330842i	$-0.107333\pi$
$864$		0			0
$865$		−	2.04161e7i		−	0.927752i
$866$	3.06923e7			1.39071
$867$		0			0
$868$		−	4.22692e6i		−	0.190425i
$869$			1.41620e6i			0.0636172i
$870$		0			0
$871$			5.75664e6i			0.257113i
$872$	3.66508e7			1.63227
$873$		0			0
$874$	−6.52472e6	+	4.54779e6i	−0.288924	+	0.201383i
$875$		−	4.55822e7i		−	2.01268i
$876$		0			0
$877$	−2.12253e7			−0.931868			−0.465934	−	0.884820i	$-0.654282\pi$
$877$	−2.12253e7			−0.931868			−0.465934	+	0.884820i	$0.654282\pi$
$878$			1.60853e7i			0.704194i
$879$		0			0
$880$	−1.39674e7			−0.608008
$881$	−1.14357e7			−0.496392			−0.248196	−	0.968710i	$-0.579838\pi$
$881$	−1.14357e7			−0.496392			−0.248196	+	0.968710i	$0.579838\pi$
$882$		0			0
$883$	−6.18808e6			−0.267088			−0.133544	−	0.991043i	$-0.542636\pi$
$883$	−6.18808e6			−0.267088			−0.133544	+	0.991043i	$0.542636\pi$
$884$	−845553.			−0.0363923
$885$		0			0
$886$	6.92929e6			0.296554
$887$			4.18950e7i			1.78794i	0.448126	+	0.893970i	$0.352091\pi$
$887$			4.18950e7i			1.78794i	−0.448126	+	0.893970i	$0.647909\pi$
$888$		0			0
$889$		−	1.97722e7i		−	0.839075i
$890$			4.34785e7i			1.83992i
$891$		0			0
$892$	−436103.			−0.0183517
$893$	−6.08532e6			−0.255361
$894$		0			0
$895$			7.50756e7i			3.13286i
$896$	1.92101e7			0.799392
$897$		0			0
$898$	2.16082e7			0.894185
$899$		−	2.44575e7i		−	1.00928i
$900$		0			0
$901$	−6.92350e6			−0.284128
$902$	1.62142e6			0.0663560
$903$		0			0
$904$		−	2.68652e7i		−	1.09338i
$905$			6.79293e7i			2.75699i
$906$		0			0
$907$			4.49675e7i			1.81502i	0.420035	+	0.907508i	$0.362018\pi$
$907$			4.49675e7i			1.81502i	−0.420035	+	0.907508i	$0.637982\pi$
$908$	−4.05101e6			−0.163060
$909$		0			0
$910$	−1.64774e7			−0.659607
$911$	−3.54093e7			−1.41358			−0.706792	−	0.707422i	$-0.749858\pi$
$911$	−3.54093e7			−1.41358			−0.706792	+	0.707422i	$0.749858\pi$
$912$		0			0
$913$	−1.10453e7			−0.438533
$914$	−5.97335e6			−0.236511
$915$		0			0
$916$		−	6.72734e6i		−	0.264914i
$917$	2.98391e7			1.17182
$918$		0			0
$919$			8.24459e6i			0.322018i	0.986953	+	0.161009i	$0.0514748\pi$
$919$			8.24459e6i			0.322018i	−0.986953	+	0.161009i	$0.948525\pi$
$920$	2.41313e7	+	3.46212e7i	0.939962	+	1.34857i
$921$		0			0
$922$	5.29314e7			2.05063
$923$			1.70718e7i			0.659590i
$924$		0			0
$925$		−	5.11072e7i		−	1.96394i
$926$			972483.i			0.0372696i
$927$		0			0
$928$	−8.21041e6			−0.312964
$929$			2.31313e7i			0.879347i	0.898158	+	0.439673i	$0.144906\pi$
$929$			2.31313e7i			0.879347i	−0.898158	+	0.439673i	$0.855094\pi$
$930$		0			0
$931$		−	4.68297e6i		−	0.177071i
$932$			8.41254e6i			0.317239i
$933$		0			0
$934$		−	1.60812e7i		−	0.603185i
$935$	−5.05468e6			−0.189088
$936$		0			0
$937$		−	1.64771e7i		−	0.613102i	−0.951854	−	0.306551i	$-0.900825\pi$
$937$		−	1.64771e7i		−	0.613102i	0.951854	−	0.306551i	$-0.0991750\pi$
$938$			1.07688e7i			0.399632i
$939$		0			0
$940$		−	8.68062e6i		−	0.320428i
$941$	−5.52339e6			−0.203344			−0.101672	−	0.994818i	$-0.532419\pi$
$941$	−5.52339e6			−0.203344			−0.101672	+	0.994818i	$0.532419\pi$
$942$		0			0
$943$	−3.42273e6	−	4.91059e6i	−0.125341	−	0.179827i
$944$		−	3.59256e7i		−	1.31212i
$945$		0			0
$946$	−979756.			−0.0355951
$947$			2.75327e7i			0.997641i	0.866705	+	0.498820i	$0.166233\pi$
$947$			2.75327e7i			0.997641i	−0.866705	+	0.498820i	$0.833767\pi$
$948$		0			0
$949$	1.23560e7			0.445362
$950$	2.53707e7			0.912060
$951$		0			0
$952$	5.88372e6			0.210407
$953$	2.47763e7			0.883698			0.441849	−	0.897089i	$-0.354323\pi$
$953$	2.47763e7			0.883698			0.441849	+	0.897089i	$0.354323\pi$
$954$		0			0
$955$	−3.87233e7			−1.37393
$956$			1.02257e7i			0.361867i
$957$		0			0
$958$			2.31107e6i			0.0813577i
$959$		−	2.92639e7i		−	1.02751i
$960$		0			0
$961$	2.31627e7			0.809061
$962$	−1.13409e7			−0.395101
$963$		0			0
$964$			3.85846e6i			0.133728i
$965$	1.89463e7			0.654948
$966$		0			0
$967$	−3.74560e7			−1.28812			−0.644059	−	0.764976i	$-0.722751\pi$
$967$	−3.74560e7			−1.28812			−0.644059	+	0.764976i	$0.722751\pi$
$968$		−	2.33810e7i		−	0.802001i
$969$		0			0
$970$	8.44014e7			2.88019
$971$	−1.91537e7			−0.651935			−0.325967	−	0.945381i	$-0.605690\pi$
$971$	−1.91537e7			−0.651935			−0.325967	+	0.945381i	$0.605690\pi$
$972$		0			0
$973$		−	7.56611e6i		−	0.256207i
$974$			2.21026e7i			0.746526i
$975$		0			0
$976$			1.64448e7i			0.552592i
$977$	−1.69586e7			−0.568400			−0.284200	−	0.958765i	$-0.591728\pi$
$977$	−1.69586e7			−0.568400			−0.284200	+	0.958765i	$0.591728\pi$
$978$		0			0
$979$	7.27459e6			0.242578
$980$	6.68018e6			0.222189
$981$		0			0
$982$	−2.12854e7			−0.704373
$983$	4.16878e7			1.37602			0.688011	−	0.725700i	$-0.258484\pi$
$983$	4.16878e7			1.37602			0.688011	+	0.725700i	$0.258484\pi$
$984$		0			0
$985$		−	4.74536e7i		−	1.55840i
$986$	−9.15224e6			−0.299802
$987$		0			0
$988$		−	984284.i		−	0.0320795i
$989$	2.06821e6	+	2.96726e6i	0.0672362	+	0.0964639i
$990$		0			0
$991$	−658077.			−0.0212859			−0.0106430	−	0.999943i	$-0.503388\pi$
$991$	−658077.			−0.0212859			−0.0106430	+	0.999943i	$0.503388\pi$
$992$		−	1.73866e7i		−	0.560964i
$993$		0			0
$994$			3.19357e7i			1.02521i
$995$		−	8.88057e7i		−	2.84370i
$996$		0			0
$997$	8.48634e6			0.270385			0.135193	−	0.990819i	$-0.456835\pi$
$997$	8.48634e6			0.270385			0.135193	+	0.990819i	$0.456835\pi$
$998$			1.53216e7i			0.486942i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.c.a.206.2	yes	40
3.2	odd	2	inner	207.6.c.a.206.39	yes	40
23.22	odd	2	inner	207.6.c.a.206.40	yes	40
69.68	even	2	inner	207.6.c.a.206.1	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.c.a.206.1	✓	40	69.68	even	2	inner
207.6.c.a.206.2	yes	40	1.1	even	1	trivial
207.6.c.a.206.39	yes	40	3.2	odd	2	inner
207.6.c.a.206.40	yes	40	23.22	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.22736i q^{2} -6.78003 q^{4} -105.915 q^{5} +86.6285i q^{7} +157.054i q^{8} +O(q^{10})\)	\(q+6.22736i q^{2} -6.78003 q^{4} -105.915 q^{5} +86.6285i q^{7} +157.054i q^{8} -659.570i q^{10} -110.356 q^{11} -288.381 q^{13} -539.467 q^{14} -1194.99 q^{16} -432.456 q^{17} -503.410i q^{19} +718.106 q^{20} -687.224i q^{22} +(-2081.31 + 1450.69i) q^{23} +8092.95 q^{25} -1795.85i q^{26} -587.344i q^{28} -3398.45i q^{29} +7196.66 q^{31} -2415.93i q^{32} -2693.06i q^{34} -9175.25i q^{35} -6315.03i q^{37} +3134.92 q^{38} -16634.3i q^{40} +2359.38i q^{41} -1425.67i q^{43} +748.215 q^{44} +(-9033.97 - 12961.1i) q^{46} -12088.2i q^{47} +9302.49 q^{49} +50397.7i q^{50} +1955.23 q^{52} +16009.7 q^{53} +11688.3 q^{55} -13605.3 q^{56} +21163.4 q^{58} +30063.4i q^{59} -13761.4i q^{61} +44816.2i q^{62} -23194.9 q^{64} +30543.8 q^{65} -19961.9i q^{67} +2932.07 q^{68} +57137.6 q^{70} -59198.6i q^{71} -42846.2 q^{73} +39325.9 q^{74} +3413.14i q^{76} -9559.95i q^{77} -12833.0i q^{79} +126567. q^{80} -14692.7 q^{82} +100089. q^{83} +45803.5 q^{85} +8878.17 q^{86} -17331.8i q^{88} -65919.5 q^{89} -24982.0i q^{91} +(14111.3 - 9835.72i) q^{92} +75277.7 q^{94} +53318.6i q^{95} +127964. i q^{97} +57930.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 600 q^{4}+O(q^{10}) \) 40 * q - 600 * q^4	\( 40 q - 600 q^{4} - 1048 q^{13} + 9728 q^{16} + 14704 q^{25} + 4640 q^{31} - 91864 q^{46} - 8192 q^{49} + 150360 q^{52} + 134592 q^{55} - 195704 q^{58} - 183416 q^{64} - 257448 q^{70} + 31088 q^{73} - 77096 q^{82} - 368760 q^{85} - 123512 q^{94}+O(q^{100}) \) 40 * q - 600 * q^4 - 1048 * q^13 + 9728 * q^16 + 14704 * q^25 + 4640 * q^31 - 91864 * q^46 - 8192 * q^49 + 150360 * q^52 + 134592 * q^55 - 195704 * q^58 - 183416 * q^64 - 257448 * q^70 + 31088 * q^73 - 77096 * q^82 - 368760 * q^85 - 123512 * q^94

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